A 2.5m long string vibrates as a 100 Hz standing wave with nodes at 1m and 1.5m from one end of the string and no points in between the two.Which harmonic is this?work out
Personally, I would like to see this with no other nodes. However, assuming there are nodes at .5m, 1m, 1.5m, 2.0m and at the neds, then 1/2 lambda is .5m, and the fundamental is 2.5m x 2=5 m
Looks like this is the fifth harmonic.
answer
To determine the harmonic of the standing wave, we need to first find the wavelength (λ) of the wave.
The distance between the two nodes (1m and 1.5m) corresponds to half a wavelength (λ/2). Therefore, the distance between the two nodes is:
d = (1.5m - 1m) = 0.5m
Since this distance corresponds to half a wavelength, we can express it as:
d = λ/2
Rearranging the equation, we can solve for the wavelength (λ):
λ = 2d
λ = 2 * 0.5m
λ = 1m
The wavelength of the wave is 1 meter.
To determine the harmonic, we can use the formula:
λ = 2L/n
where:
λ is the wavelength,
L is the length of the string, and
n is the harmonic number.
Rearranging the formula to solve for n:
n = 2L/λ
Plugging in the values:
n = 2 * 2.5m / 1m
n = 5
Therefore, this is the 5th harmonic.
To determine which harmonic this is, we need to understand the concept of harmonics in standing waves.
In a standing wave, the length of the vibrating medium (in this case, the string) is related to the wavelength of the wave. Each harmonic corresponds to a different pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) on the string.
The formula to calculate the wavelength of a standing wave on a string is:
λ = 2L/n
Where:
λ is the wavelength,
L is the length of the string, and
n is the harmonic number.
In the given question, we have the following information:
Length of the string, L = 2.5m
Distance between two nodes, L/2 = 0.5m (Nodes are located at 1m and 1.5m from one end)
We need to find the harmonic number (n) when the wavelength is half of the length of the string.
Let's substitute the values in the formula and solve for n:
0.5m = 2(2.5m)/n
Now, cross-multiply and solve for n:
n = 2(2.5m)/0.5m
n = 5
Therefore, this standing wave pattern corresponds to the 5th harmonic.
The 5th harmonic means that the string has five nodes and four antinodes. Additionally, the node at 1m and the node at 1.5m indicate that the string is vibrating in a quarter of its fundamental wavelength.
Remember, harmonic numbers start from n = 1 for the fundamental frequency, n = 2 for the second harmonic, and so on.